$\"\"$

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The curve equations are $\"\"$ and $\"\"$ .

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Two curves are said to be orthogonal trajectories when the slopes of the tangent line to both the curves is equal to $\"\"$ .

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Slope of the tangent is derivative of the curve.

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Consider $\"\"$ .

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$\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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Slope of the tangent to the curve $\"\"$ is $\"\"$ .

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$\"\"$

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Consider $\"\"$ .

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$\"\"$ .

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The curve equation is $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in above expression.

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$\"\"$

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Slope of the tangent to the curve $\"\"$ is $\"\"$ .

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$\"\"$

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If the two lines with slopes $\"\"$ and $\"\"$ are perpendicular to each other , then $\"\"$ .

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$\"\"$

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$\"\"$

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Therefore the value of $\"\"$ . $\"\"$

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The value of $\"\"$ .